As in the corresponding statement from [1]} we aim for a contradiction. Suppose that the Baum–Connes conjecture holds for \(G\)  with coefficients in \(\ell ^{\infty }(\mathbb {N},\mathcal {K})\) , but that the evaluation map is not quantitatively injective.
In this situation, we can find a pair \(r,d>0\)  such that for some \(\epsilon (0,\frac{1}{4})\)  such that for every \(d^{\prime }>d,r^{\prime }>r\)  the statement \(QI(d,d^{\prime },r,r^{\prime }\epsilon )\)  does not hold, i.e we can find a class \(x_{d^{\prime },r^{\prime }}\)  that vanishes under \(ev_{r,\epsilon ,d}\)  that never vanishes on the right hand side under \(i_{r,r^{\prime },d,d^{\prime },\epsilon }\) .
To make use of this, pick an \(\epsilon \in (0,\frac{1}{4})\)  and obtain the corresponding scales \(r\)  and \(d\) . From here, pick sequences \((d_{i})_{i}\) , \((r_{i})_{i}\)  with each \(d_{i}>d\) , \(r_{i}>r\)  that tend to infinity in \(i\) , and then obtain the collection of classes \(x_{i} \in K^{r,\epsilon }_{*}(C^{*}_{L}(P_{d}(G)\)  that witness the failure of \(QI(d,d_{i},r,r_{i},\epsilon )\)  for \(G\) .
We will wrap these up into a single class in the quantitative K-group \(K^{r,\epsilon }_{*}(C^{*}_{L}(P_{d}(G),\ell ^{\infty }(\mathbb {N},\ell ^{\infty }(G,\mathcal {K}))^{G})\)  - to do this we will make use of the the naturality of the localisation algebra with respect to coefficients - recall the isomorphism we used during the proof of Theorem REF :
\(\prod _{i}\tilde{e}_{i}: C^{*}_{L}(P_{d}(G),\ell ^{\infty }(\mathbb {N},\mathcal {K}))^{G} \rightarrow \prod _{i} C^{*}_{L}(P_{d}(G))^{G}.\) 
This was constructed from the projection maps in the coefficient algebras and composition.
The idea is now clear. We take our sequence of classes \((x_{i})_{i}\)  (which live in the \((r,\epsilon )\) -K-theory of the right hand side) and we pull them back to a single class \(z\)  in the \((r,\epsilon )\) -K-theory on the left hand side under this isomorphism. Since evaluation commutes with this isomorphism (by definition), we know that \(ev_{d,r}(z)=0\) . Pushing this class into \(K_{*}(C^{*}_{L}(G,\ell ^{\infty }(\mathbb {N},\mathcal {K}))^{G})\) , we know that actually \(i_{r,d,\epsilon }(z)=0\)  due to our assumptions on the evaluation map being an isomorphism for \(G\)  with coefficients in \(\ell ^{\infty }(\mathbb {N},\mathcal {K})\) .
We're done however, as we can now find a pair \(d^{\prime },r^{\prime }>0\)  such that \(i_{r,r^{\prime },d,d^{\prime },\epsilon }(z)\)  is 0 in \(K^{r^{\prime },\epsilon }_{*}(C^{*}_{L}(P_{d^{\prime }}(G),\ell ^{\infty }(\mathbb {N},\mathcal {K}))^{G})\) . We obtain a contraction by unpacking via the isomorphism above. After doing this, we know that \(i_{r,r^{\prime },d,d^{\prime },\epsilon }(x_{i})=0\) , however by considering the first \(d_{i}>d^{\prime }, r_{i}>r^{\prime }\)  we know that \(0=i_{r^{\prime },r_{i},d^{\prime },d_{i},\epsilon }(i_{r,r^{\prime },d,d^{\prime },\epsilon }(x_{i}))=i_{r,r_{i},d,d_{i},\epsilon }(x_{i})\ne  0\)  by construction.
